LMFDB - Embedded newform 2496.2.c.n.961.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2496,2,Mod(961,2496)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2496, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2496.961"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 2496 = 2^{6} \cdot 3 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2496.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,2,0,0,0,0,0,2,0,0,0,6,0,0,0,12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$19.9306603445$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 312)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			961.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	2496.961
Dual form			2496.2.c.n.961.2

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -2.00000i q^{5} -2.00000i q^{7} +1.00000 q^{9} +4.00000i q^{11} +(3.00000 + 2.00000i) q^{13} -2.00000i q^{15} +6.00000 q^{17} -2.00000i q^{19} -2.00000i q^{21} +4.00000 q^{23} +1.00000 q^{25} +1.00000 q^{27} -6.00000 q^{29} +2.00000i q^{31} +4.00000i q^{33} -4.00000 q^{35} +8.00000i q^{37} +(3.00000 + 2.00000i) q^{39} -6.00000i q^{41} -4.00000 q^{43} -2.00000i q^{45} -8.00000i q^{47} +3.00000 q^{49} +6.00000 q^{51} -2.00000 q^{53} +8.00000 q^{55} -2.00000i q^{57} +8.00000i q^{59} +14.0000 q^{61} -2.00000i q^{63} +(4.00000 - 6.00000i) q^{65} -14.0000i q^{67} +4.00000 q^{69} +12.0000i q^{71} -4.00000i q^{73} +1.00000 q^{75} +8.00000 q^{77} +1.00000 q^{81} -12.0000i q^{85} -6.00000 q^{87} -6.00000i q^{89} +(4.00000 - 6.00000i) q^{91} +2.00000i q^{93} -4.00000 q^{95} -12.0000i q^{97} +4.00000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + 2 q^{9} + 6 q^{13} + 12 q^{17} + 8 q^{23} + 2 q^{25} + 2 q^{27} - 12 q^{29} - 8 q^{35} + 6 q^{39} - 8 q^{43} + 6 q^{49} + 12 q^{51} - 4 q^{53} + 16 q^{55} + 28 q^{61} + 8 q^{65} + 8 q^{69}+ \cdots - 8 q^{95}+O(q^{100}) $ 2 * q + 2 * q^3 + 2 * q^9 + 6 * q^13 + 12 * q^17 + 8 * q^23 + 2 * q^25 + 2 * q^27 - 12 * q^29 - 8 * q^35 + 6 * q^39 - 8 * q^43 + 6 * q^49 + 12 * q^51 - 4 * q^53 + 16 * q^55 + 28 * q^61 + 8 * q^65 + 8 * q^69 + 2 * q^75 + 16 * q^77 + 2 * q^81 - 12 * q^87 + 8 * q^91 - 8 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2496\mathbb{Z}\right)^\times$.

$n$	$703$	$769$	$833$	$1093$
$\chi(n)$	$1$	$-1$	$1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	1.00000			0.577350
$3$	1.00000			0.577350
$4$		0			0
$5$		−	2.00000i		−	0.894427i	−0.894427	−	0.447214i	$-0.852416\pi$
$5$		−	2.00000i		−	0.894427i	0.894427	−	0.447214i	$-0.147584\pi$
$6$		0			0
$7$		−	2.00000i		−	0.755929i	−0.925820	−	0.377964i	$-0.876624\pi$
$7$		−	2.00000i		−	0.755929i	0.925820	−	0.377964i	$-0.123376\pi$
$8$		0			0
$9$	1.00000			0.333333
$10$		0			0
$11$			4.00000i			1.20605i	0.797724	+	0.603023i	$0.206037\pi$
$11$			4.00000i			1.20605i	−0.797724	+	0.603023i	$0.793963\pi$
$12$		0			0
$13$	3.00000	+	2.00000i	0.832050	+	0.554700i
$13$	3.00000	+	2.00000i	0.832050	+	0.554700i
$14$		0			0
$15$		−	2.00000i		−	0.516398i
$16$		0			0
$17$	6.00000			1.45521			0.727607	−	0.685994i	$-0.240633\pi$
$17$	6.00000			1.45521			0.727607	+	0.685994i	$0.240633\pi$
$18$		0			0
$19$		−	2.00000i		−	0.458831i	−0.973329	−	0.229416i	$-0.926318\pi$
$19$		−	2.00000i		−	0.458831i	0.973329	−	0.229416i	$-0.0736815\pi$
$20$		0			0
$21$		−	2.00000i		−	0.436436i
$22$		0			0
$23$	4.00000			0.834058			0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000			0.834058			0.417029	+	0.908893i	$0.363071\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$	1.00000			0.192450
$28$		0			0
$29$	−6.00000			−1.11417			−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000			−1.11417			−0.557086	+	0.830455i	$0.688081\pi$
$30$		0			0
$31$			2.00000i			0.359211i	0.983739	+	0.179605i	$0.0574821\pi$
$31$			2.00000i			0.359211i	−0.983739	+	0.179605i	$0.942518\pi$
$32$		0			0
$33$			4.00000i			0.696311i
$34$		0			0
$35$	−4.00000			−0.676123
$36$		0			0
$37$			8.00000i			1.31519i	0.753371	+	0.657596i	$0.228427\pi$
$37$			8.00000i			1.31519i	−0.753371	+	0.657596i	$0.771573\pi$
$38$		0			0
$39$	3.00000	+	2.00000i	0.480384	+	0.320256i
$40$		0			0
$41$		−	6.00000i		−	0.937043i	−0.883452	−	0.468521i	$-0.844787\pi$
$41$		−	6.00000i		−	0.937043i	0.883452	−	0.468521i	$-0.155213\pi$
$42$		0			0
$43$	−4.00000			−0.609994			−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000			−0.609994			−0.304997	+	0.952353i	$0.598656\pi$
$44$		0			0
$45$		−	2.00000i		−	0.298142i
$46$		0			0
$47$		−	8.00000i		−	1.16692i	−0.812142	−	0.583460i	$-0.801699\pi$
$47$		−	8.00000i		−	1.16692i	0.812142	−	0.583460i	$-0.198301\pi$
$48$		0			0
$49$	3.00000			0.428571
$50$		0			0
$51$	6.00000			0.840168
$52$		0			0
$53$	−2.00000			−0.274721			−0.137361	−	0.990521i	$-0.543862\pi$
$53$	−2.00000			−0.274721			−0.137361	+	0.990521i	$0.543862\pi$
$54$		0			0
$55$	8.00000			1.07872
$56$		0			0
$57$		−	2.00000i		−	0.264906i
$58$		0			0
$59$			8.00000i			1.04151i	0.853706	+	0.520756i	$0.174350\pi$
$59$			8.00000i			1.04151i	−0.853706	+	0.520756i	$0.825650\pi$
$60$		0			0
$61$	14.0000			1.79252			0.896258	−	0.443533i	$-0.146275\pi$
$61$	14.0000			1.79252			0.896258	+	0.443533i	$0.146275\pi$
$62$		0			0
$63$		−	2.00000i		−	0.251976i
$64$		0			0
$65$	4.00000	−	6.00000i	0.496139	−	0.744208i
$66$		0			0
$67$		−	14.0000i		−	1.71037i	−0.518321	−	0.855186i	$-0.673443\pi$
$67$		−	14.0000i		−	1.71037i	0.518321	−	0.855186i	$-0.326557\pi$
$68$		0			0
$69$	4.00000			0.481543
$70$		0			0
$71$			12.0000i			1.42414i	0.702109	+	0.712069i	$0.252242\pi$
$71$			12.0000i			1.42414i	−0.702109	+	0.712069i	$0.747758\pi$
$72$		0			0
$73$		−	4.00000i		−	0.468165i	−0.972217	−	0.234082i	$-0.924791\pi$
$73$		−	4.00000i		−	0.468165i	0.972217	−	0.234082i	$-0.0752085\pi$
$74$		0			0
$75$	1.00000			0.115470
$76$		0			0
$77$	8.00000			0.911685
$78$		0			0
$79$		0			0			−	1.00000i	$-0.5\pi$
$79$		0			0				1.00000i	$0.5\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$		0			0		1.00000			$0$
$83$		0			0		−1.00000			$\pi$
$84$		0			0
$85$		−	12.0000i		−	1.30158i
$86$		0			0
$87$	−6.00000			−0.643268
$88$		0			0
$89$		−	6.00000i		−	0.635999i	−0.948091	−	0.317999i	$-0.896989\pi$
$89$		−	6.00000i		−	0.635999i	0.948091	−	0.317999i	$-0.103011\pi$
$90$		0			0
$91$	4.00000	−	6.00000i	0.419314	−	0.628971i
$92$		0			0
$93$			2.00000i			0.207390i
$94$		0			0
$95$	−4.00000			−0.410391
$96$		0			0
$97$		−	12.0000i		−	1.21842i	−0.793011	−	0.609208i	$-0.791488\pi$
$97$		−	12.0000i		−	1.21842i	0.793011	−	0.609208i	$-0.208512\pi$
$98$		0			0
$99$			4.00000i			0.402015i
$100$		0			0
$101$	6.00000			0.597022			0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000			0.597022			0.298511	+	0.954406i	$0.403510\pi$
$102$		0			0
$103$	−8.00000			−0.788263			−0.394132	−	0.919054i	$-0.628955\pi$
$103$	−8.00000			−0.788263			−0.394132	+	0.919054i	$0.628955\pi$
$104$		0			0
$105$	−4.00000			−0.390360
$106$		0			0
$107$	−16.0000			−1.54678			−0.773389	−	0.633932i	$-0.781440\pi$
$107$	−16.0000			−1.54678			−0.773389	+	0.633932i	$0.781440\pi$
$108$		0			0
$109$		−	20.0000i		−	1.91565i	−0.287348	−	0.957826i	$-0.592774\pi$
$109$		−	20.0000i		−	1.91565i	0.287348	−	0.957826i	$-0.407226\pi$
$110$		0			0
$111$			8.00000i			0.759326i
$112$		0			0
$113$	6.00000			0.564433			0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000			0.564433			0.282216	+	0.959351i	$0.408930\pi$
$114$		0			0
$115$		−	8.00000i		−	0.746004i
$116$		0			0
$117$	3.00000	+	2.00000i	0.277350	+	0.184900i
$118$		0			0
$119$		−	12.0000i		−	1.10004i
$120$		0			0
$121$	−5.00000			−0.454545
$122$		0			0
$123$		−	6.00000i		−	0.541002i
$124$		0			0
$125$		−	12.0000i		−	1.07331i
$126$		0			0
$127$	−8.00000			−0.709885			−0.354943	−	0.934888i	$-0.615500\pi$
$127$	−8.00000			−0.709885			−0.354943	+	0.934888i	$0.615500\pi$
$128$		0			0
$129$	−4.00000			−0.352180
$130$		0			0
$131$		0			0			−	1.00000i	$-0.5\pi$
$131$		0			0				1.00000i	$0.5\pi$
$132$		0			0
$133$	−4.00000			−0.346844
$134$		0			0
$135$		−	2.00000i		−	0.172133i
$136$		0			0
$137$			14.0000i			1.19610i	0.801459	+	0.598050i	$0.204058\pi$
$137$			14.0000i			1.19610i	−0.801459	+	0.598050i	$0.795942\pi$
$138$		0			0
$139$	20.0000			1.69638			0.848189	−	0.529694i	$-0.177693\pi$
$139$	20.0000			1.69638			0.848189	+	0.529694i	$0.177693\pi$
$140$		0			0
$141$		−	8.00000i		−	0.673722i
$142$		0			0
$143$	−8.00000	+	12.0000i	−0.668994	+	1.00349i
$144$		0			0
$145$			12.0000i			0.996546i
$146$		0			0
$147$	3.00000			0.247436
$148$		0			0
$149$			2.00000i			0.163846i	0.996639	+	0.0819232i	$0.0261062\pi$
$149$			2.00000i			0.163846i	−0.996639	+	0.0819232i	$0.973894\pi$
$150$		0			0
$151$		−	14.0000i		−	1.13930i	−0.821886	−	0.569652i	$-0.807078\pi$
$151$		−	14.0000i		−	1.13930i	0.821886	−	0.569652i	$-0.192922\pi$
$152$		0			0
$153$	6.00000			0.485071
$154$		0			0
$155$	4.00000			0.321288
$156$		0			0
$157$	2.00000			0.159617			0.0798087	−	0.996810i	$-0.474569\pi$
$157$	2.00000			0.159617			0.0798087	+	0.996810i	$0.474569\pi$
$158$		0			0
$159$	−2.00000			−0.158610
$160$		0			0
$161$		−	8.00000i		−	0.630488i
$162$		0			0
$163$			22.0000i			1.72317i	0.507611	+	0.861586i	$0.330529\pi$
$163$			22.0000i			1.72317i	−0.507611	+	0.861586i	$0.669471\pi$
$164$		0			0
$165$	8.00000			0.622799
$166$		0			0
$167$			8.00000i			0.619059i	0.950890	+	0.309529i	$0.100171\pi$
$167$			8.00000i			0.619059i	−0.950890	+	0.309529i	$0.899829\pi$
$168$		0			0
$169$	5.00000	+	12.0000i	0.384615	+	0.923077i
$170$		0			0
$171$		−	2.00000i		−	0.152944i
$172$		0			0
$173$	−2.00000			−0.152057			−0.0760286	−	0.997106i	$-0.524224\pi$
$173$	−2.00000			−0.152057			−0.0760286	+	0.997106i	$0.524224\pi$
$174$		0			0
$175$		−	2.00000i		−	0.151186i
$176$		0			0
$177$			8.00000i			0.601317i
$178$		0			0
$179$	−8.00000			−0.597948			−0.298974	−	0.954261i	$-0.596644\pi$
$179$	−8.00000			−0.597948			−0.298974	+	0.954261i	$0.596644\pi$
$180$		0			0
$181$	10.0000			0.743294			0.371647	−	0.928374i	$-0.378793\pi$
$181$	10.0000			0.743294			0.371647	+	0.928374i	$0.378793\pi$
$182$		0			0
$183$	14.0000			1.03491
$184$		0			0
$185$	16.0000			1.17634
$186$		0			0
$187$			24.0000i			1.75505i
$188$		0			0
$189$		−	2.00000i		−	0.145479i
$190$		0			0
$191$	−20.0000			−1.44715			−0.723575	−	0.690246i	$-0.757502\pi$
$191$	−20.0000			−1.44715			−0.723575	+	0.690246i	$0.757502\pi$
$192$		0			0
$193$		−	16.0000i		−	1.15171i	−0.817554	−	0.575853i	$-0.804670\pi$
$193$		−	16.0000i		−	1.15171i	0.817554	−	0.575853i	$-0.195330\pi$
$194$		0			0
$195$	4.00000	−	6.00000i	0.286446	−	0.429669i
$196$		0			0
$197$			6.00000i			0.427482i	0.976890	+	0.213741i	$0.0685649\pi$
$197$			6.00000i			0.427482i	−0.976890	+	0.213741i	$0.931435\pi$
$198$		0			0
$199$		0			0			−	1.00000i	$-0.5\pi$
$199$		0			0				1.00000i	$0.5\pi$
$200$		0			0
$201$		−	14.0000i		−	0.987484i
$202$		0			0
$203$			12.0000i			0.842235i
$204$		0			0
$205$	−12.0000			−0.838116
$206$		0			0
$207$	4.00000			0.278019
$208$		0			0
$209$	8.00000			0.553372
$210$		0			0
$211$	12.0000			0.826114			0.413057	−	0.910705i	$-0.364461\pi$
$211$	12.0000			0.826114			0.413057	+	0.910705i	$0.364461\pi$
$212$		0			0
$213$			12.0000i			0.822226i
$214$		0			0
$215$			8.00000i			0.545595i
$216$		0			0
$217$	4.00000			0.271538
$218$		0			0
$219$		−	4.00000i		−	0.270295i
$220$		0			0
$221$	18.0000	+	12.0000i	1.21081	+	0.807207i
$222$		0			0
$223$			10.0000i			0.669650i	0.942280	+	0.334825i	$0.108677\pi$
$223$			10.0000i			0.669650i	−0.942280	+	0.334825i	$0.891323\pi$
$224$		0			0
$225$	1.00000			0.0666667
$226$		0			0
$227$		−	4.00000i		−	0.265489i	−0.991150	−	0.132745i	$-0.957621\pi$
$227$		−	4.00000i		−	0.265489i	0.991150	−	0.132745i	$-0.0423790\pi$
$228$		0			0
$229$		−	20.0000i		−	1.32164i	−0.750546	−	0.660819i	$-0.770209\pi$
$229$		−	20.0000i		−	1.32164i	0.750546	−	0.660819i	$-0.229791\pi$
$230$		0			0
$231$	8.00000			0.526361
$232$		0			0
$233$	−22.0000			−1.44127			−0.720634	−	0.693316i	$-0.756149\pi$
$233$	−22.0000			−1.44127			−0.720634	+	0.693316i	$0.756149\pi$
$234$		0			0
$235$	−16.0000			−1.04372
$236$		0			0
$237$		0			0
$238$		0			0
$239$			12.0000i			0.776215i	0.921614	+	0.388108i	$0.126871\pi$
$239$			12.0000i			0.776215i	−0.921614	+	0.388108i	$0.873129\pi$
$240$		0			0
$241$			4.00000i			0.257663i	0.991667	+	0.128831i	$0.0411226\pi$
$241$			4.00000i			0.257663i	−0.991667	+	0.128831i	$0.958877\pi$
$242$		0			0
$243$	1.00000			0.0641500
$244$		0			0
$245$		−	6.00000i		−	0.383326i
$246$		0			0
$247$	4.00000	−	6.00000i	0.254514	−	0.381771i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−28.0000			−1.76734			−0.883672	−	0.468106i	$-0.844936\pi$
$251$	−28.0000			−1.76734			−0.883672	+	0.468106i	$0.844936\pi$
$252$		0			0
$253$			16.0000i			1.00591i
$254$		0			0
$255$		−	12.0000i		−	0.751469i
$256$		0			0
$257$	2.00000			0.124757			0.0623783	−	0.998053i	$-0.480131\pi$
$257$	2.00000			0.124757			0.0623783	+	0.998053i	$0.480131\pi$
$258$		0			0
$259$	16.0000			0.994192
$260$		0			0
$261$	−6.00000			−0.371391
$262$		0			0
$263$	24.0000			1.47990			0.739952	−	0.672660i	$-0.234848\pi$
$263$	24.0000			1.47990			0.739952	+	0.672660i	$0.234848\pi$
$264$		0			0
$265$			4.00000i			0.245718i
$266$		0			0
$267$		−	6.00000i		−	0.367194i
$268$		0			0
$269$	10.0000			0.609711			0.304855	−	0.952399i	$-0.401392\pi$
$269$	10.0000			0.609711			0.304855	+	0.952399i	$0.401392\pi$
$270$		0			0
$271$			22.0000i			1.33640i	0.743980	+	0.668202i	$0.232936\pi$
$271$			22.0000i			1.33640i	−0.743980	+	0.668202i	$0.767064\pi$
$272$		0			0
$273$	4.00000	−	6.00000i	0.242091	−	0.363137i
$274$		0			0
$275$			4.00000i			0.241209i
$276$		0			0
$277$	−14.0000			−0.841178			−0.420589	−	0.907251i	$-0.638177\pi$
$277$	−14.0000			−0.841178			−0.420589	+	0.907251i	$0.638177\pi$
$278$		0			0
$279$			2.00000i			0.119737i
$280$		0			0
$281$			22.0000i			1.31241i	0.754583	+	0.656205i	$0.227839\pi$
$281$			22.0000i			1.31241i	−0.754583	+	0.656205i	$0.772161\pi$
$282$		0			0
$283$	−28.0000			−1.66443			−0.832214	−	0.554455i	$-0.812927\pi$
$283$	−28.0000			−1.66443			−0.832214	+	0.554455i	$0.812927\pi$
$284$		0			0
$285$	−4.00000			−0.236940
$286$		0			0
$287$	−12.0000			−0.708338
$288$		0			0
$289$	19.0000			1.11765
$290$		0			0
$291$		−	12.0000i		−	0.703452i
$292$		0			0
$293$			30.0000i			1.75262i	0.481749	+	0.876309i	$0.340002\pi$
$293$			30.0000i			1.75262i	−0.481749	+	0.876309i	$0.659998\pi$
$294$		0			0
$295$	16.0000			0.931556
$296$		0			0
$297$			4.00000i			0.232104i
$298$		0			0
$299$	12.0000	+	8.00000i	0.693978	+	0.462652i
$300$		0			0
$301$			8.00000i			0.461112i
$302$		0			0
$303$	6.00000			0.344691
$304$		0			0
$305$		−	28.0000i		−	1.60328i
$306$		0			0
$307$		−	30.0000i		−	1.71219i	−0.516818	−	0.856095i	$-0.672884\pi$
$307$		−	30.0000i		−	1.71219i	0.516818	−	0.856095i	$-0.327116\pi$
$308$		0			0
$309$	−8.00000			−0.455104
$310$		0			0
$311$	−12.0000			−0.680458			−0.340229	−	0.940343i	$-0.610505\pi$
$311$	−12.0000			−0.680458			−0.340229	+	0.940343i	$0.610505\pi$
$312$		0			0
$313$	22.0000			1.24351			0.621757	−	0.783210i	$-0.286419\pi$
$313$	22.0000			1.24351			0.621757	+	0.783210i	$0.286419\pi$
$314$		0			0
$315$	−4.00000			−0.225374
$316$		0			0
$317$		−	18.0000i		−	1.01098i	−0.862832	−	0.505490i	$-0.831312\pi$
$317$		−	18.0000i		−	1.01098i	0.862832	−	0.505490i	$-0.168688\pi$
$318$		0			0
$319$		−	24.0000i		−	1.34374i
$320$		0			0
$321$	−16.0000			−0.893033
$322$		0			0
$323$		−	12.0000i		−	0.667698i
$324$		0			0
$325$	3.00000	+	2.00000i	0.166410	+	0.110940i
$326$		0			0
$327$		−	20.0000i		−	1.10600i
$328$		0			0
$329$	−16.0000			−0.882109
$330$		0			0
$331$		−	6.00000i		−	0.329790i	−0.986311	−	0.164895i	$-0.947272\pi$
$331$		−	6.00000i		−	0.329790i	0.986311	−	0.164895i	$-0.0527285\pi$
$332$		0			0
$333$			8.00000i			0.438397i
$334$		0			0
$335$	−28.0000			−1.52980
$336$		0			0
$337$	−34.0000			−1.85210			−0.926049	−	0.377403i	$-0.876817\pi$
$337$	−34.0000			−1.85210			−0.926049	+	0.377403i	$0.876817\pi$
$338$		0			0
$339$	6.00000			0.325875
$340$		0			0
$341$	−8.00000			−0.433224
$342$		0			0
$343$		−	20.0000i		−	1.07990i
$344$		0			0
$345$		−	8.00000i		−	0.430706i
$346$		0			0
$347$	−28.0000			−1.50312			−0.751559	−	0.659665i	$-0.770698\pi$
$347$	−28.0000			−1.50312			−0.751559	+	0.659665i	$0.770698\pi$
$348$		0			0
$349$			32.0000i			1.71292i	0.516213	+	0.856460i	$0.327341\pi$
$349$			32.0000i			1.71292i	−0.516213	+	0.856460i	$0.672659\pi$
$350$		0			0
$351$	3.00000	+	2.00000i	0.160128	+	0.106752i
$352$		0			0
$353$			10.0000i			0.532246i	0.963939	+	0.266123i	$0.0857428\pi$
$353$			10.0000i			0.532246i	−0.963939	+	0.266123i	$0.914257\pi$
$354$		0			0
$355$	24.0000			1.27379
$356$		0			0
$357$		−	12.0000i		−	0.635107i
$358$		0			0
$359$		−	8.00000i		−	0.422224i	−0.977462	−	0.211112i	$-0.932292\pi$
$359$		−	8.00000i		−	0.422224i	0.977462	−	0.211112i	$-0.0677085\pi$
$360$		0			0
$361$	15.0000			0.789474
$362$		0			0
$363$	−5.00000			−0.262432
$364$		0			0
$365$	−8.00000			−0.418739
$366$		0			0
$367$	32.0000			1.67039			0.835193	−	0.549957i	$-0.185356\pi$
$367$	32.0000			1.67039			0.835193	+	0.549957i	$0.185356\pi$
$368$		0			0
$369$		−	6.00000i		−	0.312348i
$370$		0			0
$371$			4.00000i			0.207670i
$372$		0			0
$373$	6.00000			0.310668			0.155334	−	0.987862i	$-0.450355\pi$
$373$	6.00000			0.310668			0.155334	+	0.987862i	$0.450355\pi$
$374$		0			0
$375$		−	12.0000i		−	0.619677i
$376$		0			0
$377$	−18.0000	−	12.0000i	−0.927047	−	0.618031i
$378$		0			0
$379$			22.0000i			1.13006i	0.825069	+	0.565032i	$0.191136\pi$
$379$			22.0000i			1.13006i	−0.825069	+	0.565032i	$0.808864\pi$
$380$		0			0
$381$	−8.00000			−0.409852
$382$		0			0
$383$			16.0000i			0.817562i	0.912633	+	0.408781i	$0.134046\pi$
$383$			16.0000i			0.817562i	−0.912633	+	0.408781i	$0.865954\pi$
$384$		0			0
$385$		−	16.0000i		−	0.815436i
$386$		0			0
$387$	−4.00000			−0.203331
$388$		0			0
$389$	−30.0000			−1.52106			−0.760530	−	0.649303i	$-0.775061\pi$
$389$	−30.0000			−1.52106			−0.760530	+	0.649303i	$0.775061\pi$
$390$		0			0
$391$	24.0000			1.21373
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$			8.00000i			0.401508i	0.979642	+	0.200754i	$0.0643393\pi$
$397$			8.00000i			0.401508i	−0.979642	+	0.200754i	$0.935661\pi$
$398$		0			0
$399$	−4.00000			−0.200250
$400$		0			0
$401$			34.0000i			1.69788i	0.528490	+	0.848939i	$0.322758\pi$
$401$			34.0000i			1.69788i	−0.528490	+	0.848939i	$0.677242\pi$
$402$		0			0
$403$	−4.00000	+	6.00000i	−0.199254	+	0.298881i
$404$		0			0
$405$		−	2.00000i		−	0.0993808i
$406$		0			0
$407$	−32.0000			−1.58618
$408$		0			0
$409$			20.0000i			0.988936i	0.869196	+	0.494468i	$0.164637\pi$
$409$			20.0000i			0.988936i	−0.869196	+	0.494468i	$0.835363\pi$
$410$		0			0
$411$			14.0000i			0.690569i
$412$		0			0
$413$	16.0000			0.787309
$414$		0			0
$415$		0			0
$416$		0			0
$417$	20.0000			0.979404
$418$		0			0
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$		0			0
$421$		−	4.00000i		−	0.194948i	−0.995238	−	0.0974740i	$-0.968924\pi$
$421$		−	4.00000i		−	0.194948i	0.995238	−	0.0974740i	$-0.0310763\pi$
$422$		0			0
$423$		−	8.00000i		−	0.388973i
$424$		0			0
$425$	6.00000			0.291043
$426$		0			0
$427$		−	28.0000i		−	1.35501i
$428$		0			0
$429$	−8.00000	+	12.0000i	−0.386244	+	0.579365i
$430$		0			0
$431$			16.0000i			0.770693i	0.922772	+	0.385346i	$0.125918\pi$
$431$			16.0000i			0.770693i	−0.922772	+	0.385346i	$0.874082\pi$
$432$		0			0
$433$	−26.0000			−1.24948			−0.624740	−	0.780833i	$-0.714795\pi$
$433$	−26.0000			−1.24948			−0.624740	+	0.780833i	$0.714795\pi$
$434$		0			0
$435$			12.0000i			0.575356i
$436$		0			0
$437$		−	8.00000i		−	0.382692i
$438$		0			0
$439$	24.0000			1.14546			0.572729	−	0.819745i	$-0.305885\pi$
$439$	24.0000			1.14546			0.572729	+	0.819745i	$0.305885\pi$
$440$		0			0
$441$	3.00000			0.142857
$442$		0			0
$443$	−24.0000			−1.14027			−0.570137	−	0.821549i	$-0.693110\pi$
$443$	−24.0000			−1.14027			−0.570137	+	0.821549i	$0.693110\pi$
$444$		0			0
$445$	−12.0000			−0.568855
$446$		0			0
$447$			2.00000i			0.0945968i
$448$		0			0
$449$			26.0000i			1.22702i	0.789689	+	0.613508i	$0.210242\pi$
$449$			26.0000i			1.22702i	−0.789689	+	0.613508i	$0.789758\pi$
$450$		0			0
$451$	24.0000			1.13012
$452$		0			0
$453$		−	14.0000i		−	0.657777i
$454$		0			0
$455$	−12.0000	−	8.00000i	−0.562569	−	0.375046i
$456$		0			0
$457$		−	20.0000i		−	0.935561i	−0.883845	−	0.467780i	$-0.845054\pi$
$457$		−	20.0000i		−	0.935561i	0.883845	−	0.467780i	$-0.154946\pi$
$458$		0			0
$459$	6.00000			0.280056
$460$		0			0
$461$		−	18.0000i		−	0.838344i	−0.907907	−	0.419172i	$-0.862320\pi$
$461$		−	18.0000i		−	0.838344i	0.907907	−	0.419172i	$-0.137680\pi$
$462$		0			0
$463$			22.0000i			1.02243i	0.859454	+	0.511213i	$0.170804\pi$
$463$			22.0000i			1.02243i	−0.859454	+	0.511213i	$0.829196\pi$
$464$		0			0
$465$	4.00000			0.185496
$466$		0			0
$467$	36.0000			1.66588			0.832941	−	0.553362i	$-0.186655\pi$
$467$	36.0000			1.66588			0.832941	+	0.553362i	$0.186655\pi$
$468$		0			0
$469$	−28.0000			−1.29292
$470$		0			0
$471$	2.00000			0.0921551
$472$		0			0
$473$		−	16.0000i		−	0.735681i
$474$		0			0
$475$		−	2.00000i		−	0.0917663i
$476$		0			0
$477$	−2.00000			−0.0915737
$478$		0			0
$479$		−	28.0000i		−	1.27935i	−0.768644	−	0.639676i	$-0.779068\pi$
$479$		−	28.0000i		−	1.27935i	0.768644	−	0.639676i	$-0.220932\pi$
$480$		0			0
$481$	−16.0000	+	24.0000i	−0.729537	+	1.09431i
$482$		0			0
$483$		−	8.00000i		−	0.364013i
$484$		0			0
$485$	−24.0000			−1.08978
$486$		0			0
$487$		−	6.00000i		−	0.271886i	−0.990717	−	0.135943i	$-0.956594\pi$
$487$		−	6.00000i		−	0.271886i	0.990717	−	0.135943i	$-0.0434064\pi$
$488$		0			0
$489$			22.0000i			0.994874i
$490$		0			0
$491$	36.0000			1.62466			0.812329	−	0.583200i	$-0.198200\pi$
$491$	36.0000			1.62466			0.812329	+	0.583200i	$0.198200\pi$
$492$		0			0
$493$	−36.0000			−1.62136
$494$		0			0
$495$	8.00000			0.359573
$496$		0			0
$497$	24.0000			1.07655
$498$		0			0
$499$			18.0000i			0.805791i	0.915246	+	0.402895i	$0.131996\pi$
$499$			18.0000i			0.805791i	−0.915246	+	0.402895i	$0.868004\pi$
$500$		0			0
$501$			8.00000i			0.357414i
$502$		0			0
$503$	8.00000			0.356702			0.178351	−	0.983967i	$-0.442924\pi$
$503$	8.00000			0.356702			0.178351	+	0.983967i	$0.442924\pi$
$504$		0			0
$505$		−	12.0000i		−	0.533993i
$506$		0			0
$507$	5.00000	+	12.0000i	0.222058	+	0.532939i
$508$		0			0
$509$		−	26.0000i		−	1.15243i	−0.817298	−	0.576215i	$-0.804529\pi$
$509$		−	26.0000i		−	1.15243i	0.817298	−	0.576215i	$-0.195471\pi$
$510$		0			0
$511$	−8.00000			−0.353899
$512$		0			0
$513$		−	2.00000i		−	0.0883022i
$514$		0			0
$515$			16.0000i			0.705044i
$516$		0			0
$517$	32.0000			1.40736
$518$		0			0
$519$	−2.00000			−0.0877903
$520$		0			0
$521$	−42.0000			−1.84005			−0.920027	−	0.391856i	$-0.871833\pi$
$521$	−42.0000			−1.84005			−0.920027	+	0.391856i	$0.871833\pi$
$522$		0			0
$523$	−4.00000			−0.174908			−0.0874539	−	0.996169i	$-0.527873\pi$
$523$	−4.00000			−0.174908			−0.0874539	+	0.996169i	$0.527873\pi$
$524$		0			0
$525$		−	2.00000i		−	0.0872872i
$526$		0			0
$527$			12.0000i			0.522728i
$528$		0			0
$529$	−7.00000			−0.304348
$530$		0			0
$531$			8.00000i			0.347170i
$532$		0			0
$533$	12.0000	−	18.0000i	0.519778	−	0.779667i
$534$		0			0
$535$			32.0000i			1.38348i
$536$		0			0
$537$	−8.00000			−0.345225
$538$		0			0
$539$			12.0000i			0.516877i
$540$		0			0
$541$			20.0000i			0.859867i	0.902861	+	0.429934i	$0.141463\pi$
$541$			20.0000i			0.859867i	−0.902861	+	0.429934i	$0.858537\pi$
$542$		0			0
$543$	10.0000			0.429141
$544$		0			0
$545$	−40.0000			−1.71341
$546$		0			0
$547$	4.00000			0.171028			0.0855138	−	0.996337i	$-0.472747\pi$
$547$	4.00000			0.171028			0.0855138	+	0.996337i	$0.472747\pi$
$548$		0			0
$549$	14.0000			0.597505
$550$		0			0
$551$			12.0000i			0.511217i
$552$		0			0
$553$		0			0
$554$		0			0
$555$	16.0000			0.679162
$556$		0			0
$557$		−	2.00000i		−	0.0847427i	−0.999102	−	0.0423714i	$-0.986509\pi$
$557$		−	2.00000i		−	0.0847427i	0.999102	−	0.0423714i	$-0.0134913\pi$
$558$		0			0
$559$	−12.0000	−	8.00000i	−0.507546	−	0.338364i
$560$		0			0
$561$			24.0000i			1.01328i
$562$		0			0
$563$	−8.00000			−0.337160			−0.168580	−	0.985688i	$-0.553918\pi$
$563$	−8.00000			−0.337160			−0.168580	+	0.985688i	$0.553918\pi$
$564$		0			0
$565$		−	12.0000i		−	0.504844i
$566$		0			0
$567$		−	2.00000i		−	0.0839921i
$568$		0			0
$569$	−6.00000			−0.251533			−0.125767	−	0.992060i	$-0.540139\pi$
$569$	−6.00000			−0.251533			−0.125767	+	0.992060i	$0.540139\pi$
$570$		0			0
$571$	4.00000			0.167395			0.0836974	−	0.996491i	$-0.473327\pi$
$571$	4.00000			0.167395			0.0836974	+	0.996491i	$0.473327\pi$
$572$		0			0
$573$	−20.0000			−0.835512
$574$		0			0
$575$	4.00000			0.166812
$576$		0			0
$577$			8.00000i			0.333044i	0.986038	+	0.166522i	$0.0532537\pi$
$577$			8.00000i			0.333044i	−0.986038	+	0.166522i	$0.946746\pi$
$578$		0			0
$579$		−	16.0000i		−	0.664937i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		−	8.00000i		−	0.331326i
$584$		0			0
$585$	4.00000	−	6.00000i	0.165380	−	0.248069i
$586$		0			0
$587$		0			0		1.00000			$0$
$587$		0			0		−1.00000			$\pi$
$588$		0			0
$589$	4.00000			0.164817
$590$		0			0
$591$			6.00000i			0.246807i
$592$		0			0
$593$		−	22.0000i		−	0.903432i	−0.892162	−	0.451716i	$-0.850812\pi$
$593$		−	22.0000i		−	0.903432i	0.892162	−	0.451716i	$-0.149188\pi$
$594$		0			0
$595$	−24.0000			−0.983904
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$		0			0
$601$	−26.0000			−1.06056			−0.530281	−	0.847822i	$-0.677914\pi$
$601$	−26.0000			−1.06056			−0.530281	+	0.847822i	$0.677914\pi$
$602$		0			0
$603$		−	14.0000i		−	0.570124i
$604$		0			0
$605$			10.0000i			0.406558i
$606$		0			0
$607$	−40.0000			−1.62355			−0.811775	−	0.583970i	$-0.801498\pi$
$607$	−40.0000			−1.62355			−0.811775	+	0.583970i	$0.801498\pi$
$608$		0			0
$609$			12.0000i			0.486265i
$610$		0			0
$611$	16.0000	−	24.0000i	0.647291	−	0.970936i
$612$		0			0
$613$			32.0000i			1.29247i	0.763139	+	0.646234i	$0.223657\pi$
$613$			32.0000i			1.29247i	−0.763139	+	0.646234i	$0.776343\pi$
$614$		0			0
$615$	−12.0000			−0.483887
$616$		0			0
$617$			42.0000i			1.69086i	0.534089	+	0.845428i	$0.320655\pi$
$617$			42.0000i			1.69086i	−0.534089	+	0.845428i	$0.679345\pi$
$618$		0			0
$619$		−	38.0000i		−	1.52735i	−0.645601	−	0.763674i	$-0.723393\pi$
$619$		−	38.0000i		−	1.52735i	0.645601	−	0.763674i	$-0.276607\pi$
$620$		0			0
$621$	4.00000			0.160514
$622$		0			0
$623$	−12.0000			−0.480770
$624$		0			0
$625$	−19.0000			−0.760000
$626$		0			0
$627$	8.00000			0.319489
$628$		0			0
$629$			48.0000i			1.91389i
$630$		0			0
$631$			2.00000i			0.0796187i	0.999207	+	0.0398094i	$0.0126751\pi$
$631$			2.00000i			0.0796187i	−0.999207	+	0.0398094i	$0.987325\pi$
$632$		0			0
$633$	12.0000			0.476957
$634$		0			0
$635$			16.0000i			0.634941i
$636$		0			0
$637$	9.00000	+	6.00000i	0.356593	+	0.237729i
$638$		0			0
$639$			12.0000i			0.474713i
$640$		0			0
$641$	−26.0000			−1.02694			−0.513469	−	0.858108i	$-0.671640\pi$
$641$	−26.0000			−1.02694			−0.513469	+	0.858108i	$0.671640\pi$
$642$		0			0
$643$		−	22.0000i		−	0.867595i	−0.901010	−	0.433798i	$-0.857173\pi$
$643$		−	22.0000i		−	0.867595i	0.901010	−	0.433798i	$-0.142827\pi$
$644$		0			0
$645$			8.00000i			0.315000i
$646$		0			0
$647$		0			0			−	1.00000i	$-0.5\pi$
$647$		0			0				1.00000i	$0.5\pi$
$648$		0			0
$649$	−32.0000			−1.25611
$650$		0			0
$651$	4.00000			0.156772
$652$		0			0
$653$	−22.0000			−0.860927			−0.430463	−	0.902608i	$-0.641650\pi$
$653$	−22.0000			−0.860927			−0.430463	+	0.902608i	$0.641650\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		−	4.00000i		−	0.156055i
$658$		0			0
$659$	−12.0000			−0.467454			−0.233727	−	0.972302i	$-0.575092\pi$
$659$	−12.0000			−0.467454			−0.233727	+	0.972302i	$0.575092\pi$
$660$		0			0
$661$			8.00000i			0.311164i	0.987823	+	0.155582i	$0.0497253\pi$
$661$			8.00000i			0.311164i	−0.987823	+	0.155582i	$0.950275\pi$
$662$		0			0
$663$	18.0000	+	12.0000i	0.699062	+	0.466041i
$664$		0			0
$665$			8.00000i			0.310227i
$666$		0			0
$667$	−24.0000			−0.929284
$668$		0			0
$669$			10.0000i			0.386622i
$670$		0			0
$671$			56.0000i			2.16186i
$672$		0			0
$673$	26.0000			1.00223			0.501113	−	0.865382i	$-0.332924\pi$
$673$	26.0000			1.00223			0.501113	+	0.865382i	$0.332924\pi$
$674$		0			0
$675$	1.00000			0.0384900
$676$		0			0
$677$	6.00000			0.230599			0.115299	−	0.993331i	$-0.463217\pi$
$677$	6.00000			0.230599			0.115299	+	0.993331i	$0.463217\pi$
$678$		0			0
$679$	−24.0000			−0.921035
$680$		0			0
$681$		−	4.00000i		−	0.153280i
$682$		0			0
$683$		−	36.0000i		−	1.37750i	−0.724998	−	0.688751i	$-0.758159\pi$
$683$		−	36.0000i		−	1.37750i	0.724998	−	0.688751i	$-0.241841\pi$
$684$		0			0
$685$	28.0000			1.06983
$686$		0			0
$687$		−	20.0000i		−	0.763048i
$688$		0			0
$689$	−6.00000	−	4.00000i	−0.228582	−	0.152388i
$690$		0			0
$691$			10.0000i			0.380418i	0.981744	+	0.190209i	$0.0609166\pi$
$691$			10.0000i			0.380418i	−0.981744	+	0.190209i	$0.939083\pi$
$692$		0			0
$693$	8.00000			0.303895
$694$		0			0
$695$		−	40.0000i		−	1.51729i
$696$		0			0
$697$		−	36.0000i		−	1.36360i
$698$		0			0
$699$	−22.0000			−0.832116
$700$		0			0
$701$	10.0000			0.377695			0.188847	−	0.982006i	$-0.439525\pi$
$701$	10.0000			0.377695			0.188847	+	0.982006i	$0.439525\pi$
$702$		0			0
$703$	16.0000			0.603451
$704$		0			0
$705$	−16.0000			−0.602595
$706$		0			0
$707$		−	12.0000i		−	0.451306i
$708$		0			0
$709$		−	44.0000i		−	1.65245i	−0.563337	−	0.826227i	$-0.690483\pi$
$709$		−	44.0000i		−	1.65245i	0.563337	−	0.826227i	$-0.309517\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			8.00000i			0.299602i
$714$		0			0
$715$	24.0000	+	16.0000i	0.897549	+	0.598366i
$716$		0			0
$717$			12.0000i			0.448148i
$718$		0			0
$719$	4.00000			0.149175			0.0745874	−	0.997214i	$-0.476236\pi$
$719$	4.00000			0.149175			0.0745874	+	0.997214i	$0.476236\pi$
$720$		0			0
$721$			16.0000i			0.595871i
$722$		0			0
$723$			4.00000i			0.148762i
$724$		0			0
$725$	−6.00000			−0.222834
$726$		0			0
$727$	24.0000			0.890111			0.445055	−	0.895503i	$-0.353184\pi$
$727$	24.0000			0.890111			0.445055	+	0.895503i	$0.353184\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$	−24.0000			−0.887672
$732$		0			0
$733$		−	20.0000i		−	0.738717i	−0.929287	−	0.369358i	$-0.879577\pi$
$733$		−	20.0000i		−	0.738717i	0.929287	−	0.369358i	$-0.120423\pi$
$734$		0			0
$735$		−	6.00000i		−	0.221313i
$736$		0			0
$737$	56.0000			2.06279
$738$		0			0
$739$			26.0000i			0.956425i	0.878244	+	0.478213i	$0.158715\pi$
$739$			26.0000i			0.956425i	−0.878244	+	0.478213i	$0.841285\pi$
$740$		0			0
$741$	4.00000	−	6.00000i	0.146944	−	0.220416i
$742$		0			0
$743$		−	4.00000i		−	0.146746i	−0.997305	−	0.0733729i	$-0.976624\pi$
$743$		−	4.00000i		−	0.146746i	0.997305	−	0.0733729i	$-0.0233763\pi$
$744$		0			0
$745$	4.00000			0.146549
$746$		0			0
$747$		0			0
$748$		0			0
$749$			32.0000i			1.16925i
$750$		0			0
$751$	−40.0000			−1.45962			−0.729810	−	0.683650i	$-0.760392\pi$
$751$	−40.0000			−1.45962			−0.729810	+	0.683650i	$0.760392\pi$
$752$		0			0
$753$	−28.0000			−1.02038
$754$		0			0
$755$	−28.0000			−1.01902
$756$		0			0
$757$	22.0000			0.799604			0.399802	−	0.916602i	$-0.369079\pi$
$757$	22.0000			0.799604			0.399802	+	0.916602i	$0.369079\pi$
$758$		0			0
$759$			16.0000i			0.580763i
$760$		0			0
$761$		−	18.0000i		−	0.652499i	−0.945284	−	0.326250i	$-0.894215\pi$
$761$		−	18.0000i		−	0.652499i	0.945284	−	0.326250i	$-0.105785\pi$
$762$		0			0
$763$	−40.0000			−1.44810
$764$		0			0
$765$		−	12.0000i		−	0.433861i
$766$		0			0
$767$	−16.0000	+	24.0000i	−0.577727	+	0.866590i
$768$		0			0
$769$			24.0000i			0.865462i	0.901523	+	0.432731i	$0.142450\pi$
$769$			24.0000i			0.865462i	−0.901523	+	0.432731i	$0.857550\pi$
$770$		0			0
$771$	2.00000			0.0720282
$772$		0			0
$773$			26.0000i			0.935155i	0.883952	+	0.467578i	$0.154873\pi$
$773$			26.0000i			0.935155i	−0.883952	+	0.467578i	$0.845127\pi$
$774$		0			0
$775$			2.00000i			0.0718421i
$776$		0			0
$777$	16.0000			0.573997
$778$		0			0
$779$	−12.0000			−0.429945
$780$		0			0
$781$	−48.0000			−1.71758
$782$		0			0
$783$	−6.00000			−0.214423
$784$		0			0
$785$		−	4.00000i		−	0.142766i
$786$		0			0
$787$		−	38.0000i		−	1.35455i	−0.735728	−	0.677277i	$-0.763160\pi$
$787$		−	38.0000i		−	1.35455i	0.735728	−	0.677277i	$-0.236840\pi$
$788$		0			0
$789$	24.0000			0.854423
$790$		0			0
$791$		−	12.0000i		−	0.426671i
$792$		0			0
$793$	42.0000	+	28.0000i	1.49146	+	0.994309i
$794$		0			0
$795$			4.00000i			0.141865i
$796$		0			0
$797$	−14.0000			−0.495905			−0.247953	−	0.968772i	$-0.579758\pi$
$797$	−14.0000			−0.495905			−0.247953	+	0.968772i	$0.579758\pi$
$798$		0			0
$799$		−	48.0000i		−	1.69812i
$800$		0			0
$801$		−	6.00000i		−	0.212000i
$802$		0			0
$803$	16.0000			0.564628
$804$		0			0
$805$	−16.0000			−0.563926
$806$		0			0
$807$	10.0000			0.352017
$808$		0			0
$809$	6.00000			0.210949			0.105474	−	0.994422i	$-0.466364\pi$
$809$	6.00000			0.210949			0.105474	+	0.994422i	$0.466364\pi$
$810$		0			0
$811$		−	2.00000i		−	0.0702295i	−0.999383	−	0.0351147i	$-0.988820\pi$
$811$		−	2.00000i		−	0.0702295i	0.999383	−	0.0351147i	$-0.0111797\pi$
$812$		0			0
$813$			22.0000i			0.771574i
$814$		0			0
$815$	44.0000			1.54125
$816$		0			0
$817$			8.00000i			0.279885i
$818$		0			0
$819$	4.00000	−	6.00000i	0.139771	−	0.209657i
$820$		0			0
$821$			2.00000i			0.0698005i	0.999391	+	0.0349002i	$0.0111113\pi$
$821$			2.00000i			0.0698005i	−0.999391	+	0.0349002i	$0.988889\pi$
$822$		0			0
$823$		0			0			−	1.00000i	$-0.5\pi$
$823$		0			0				1.00000i	$0.5\pi$
$824$		0			0
$825$			4.00000i			0.139262i
$826$		0			0
$827$			52.0000i			1.80822i	0.427303	+	0.904109i	$0.359464\pi$
$827$			52.0000i			1.80822i	−0.427303	+	0.904109i	$0.640536\pi$
$828$		0			0
$829$	−34.0000			−1.18087			−0.590434	−	0.807086i	$-0.701044\pi$
$829$	−34.0000			−1.18087			−0.590434	+	0.807086i	$0.701044\pi$
$830$		0			0
$831$	−14.0000			−0.485655
$832$		0			0
$833$	18.0000			0.623663
$834$		0			0
$835$	16.0000			0.553703
$836$		0			0
$837$			2.00000i			0.0691301i
$838$		0			0
$839$			12.0000i			0.414286i	0.978311	+	0.207143i	$0.0664165\pi$
$839$			12.0000i			0.414286i	−0.978311	+	0.207143i	$0.933583\pi$
$840$		0			0
$841$	7.00000			0.241379
$842$		0			0
$843$			22.0000i			0.757720i
$844$		0			0
$845$	24.0000	−	10.0000i	0.825625	−	0.344010i
$846$		0			0
$847$			10.0000i			0.343604i
$848$		0			0
$849$	−28.0000			−0.960958
$850$		0			0
$851$			32.0000i			1.09695i
$852$		0			0
$853$			40.0000i			1.36957i	0.728743	+	0.684787i	$0.240105\pi$
$853$			40.0000i			1.36957i	−0.728743	+	0.684787i	$0.759895\pi$
$854$		0			0
$855$	−4.00000			−0.136797
$856$		0			0
$857$	10.0000			0.341593			0.170797	−	0.985306i	$-0.445366\pi$
$857$	10.0000			0.341593			0.170797	+	0.985306i	$0.445366\pi$
$858$		0			0
$859$	28.0000			0.955348			0.477674	−	0.878537i	$-0.341480\pi$
$859$	28.0000			0.955348			0.477674	+	0.878537i	$0.341480\pi$
$860$		0			0
$861$	−12.0000			−0.408959
$862$		0			0
$863$		−	24.0000i		−	0.816970i	−0.912765	−	0.408485i	$-0.866057\pi$
$863$		−	24.0000i		−	0.816970i	0.912765	−	0.408485i	$-0.133943\pi$
$864$		0			0
$865$			4.00000i			0.136004i
$866$		0			0
$867$	19.0000			0.645274
$868$		0			0
$869$		0			0
$870$		0			0
$871$	28.0000	−	42.0000i	0.948744	−	1.42312i
$872$		0			0
$873$		−	12.0000i		−	0.406138i
$874$		0			0
$875$	−24.0000			−0.811348
$876$		0			0
$877$		−	40.0000i		−	1.35070i	−0.737496	−	0.675352i	$-0.763992\pi$
$877$		−	40.0000i		−	1.35070i	0.737496	−	0.675352i	$-0.236008\pi$
$878$		0			0
$879$			30.0000i			1.01187i
$880$		0			0
$881$	10.0000			0.336909			0.168454	−	0.985709i	$-0.446122\pi$
$881$	10.0000			0.336909			0.168454	+	0.985709i	$0.446122\pi$
$882$		0			0
$883$	−20.0000			−0.673054			−0.336527	−	0.941674i	$-0.609252\pi$
$883$	−20.0000			−0.673054			−0.336527	+	0.941674i	$0.609252\pi$
$884$		0			0
$885$	16.0000			0.537834
$886$		0			0
$887$	20.0000			0.671534			0.335767	−	0.941945i	$-0.391004\pi$
$887$	20.0000			0.671534			0.335767	+	0.941945i	$0.391004\pi$
$888$		0			0
$889$			16.0000i			0.536623i
$890$		0			0
$891$			4.00000i			0.134005i
$892$		0			0
$893$	−16.0000			−0.535420
$894$		0			0
$895$			16.0000i			0.534821i
$896$		0			0
$897$	12.0000	+	8.00000i	0.400668	+	0.267112i
$898$		0			0
$899$		−	12.0000i		−	0.400222i
$900$		0			0
$901$	−12.0000			−0.399778
$902$		0			0
$903$			8.00000i			0.266223i
$904$		0			0
$905$		−	20.0000i		−	0.664822i
$906$		0			0
$907$	12.0000			0.398453			0.199227	−	0.979953i	$-0.436157\pi$
$907$	12.0000			0.398453			0.199227	+	0.979953i	$0.436157\pi$
$908$		0			0
$909$	6.00000			0.199007
$910$		0			0
$911$	−36.0000			−1.19273			−0.596367	−	0.802712i	$-0.703390\pi$
$911$	−36.0000			−1.19273			−0.596367	+	0.802712i	$0.703390\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		−	28.0000i		−	0.925651i
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−48.0000			−1.58337			−0.791687	−	0.610927i	$-0.790797\pi$
$919$	−48.0000			−1.58337			−0.791687	+	0.610927i	$0.790797\pi$
$920$		0			0
$921$		−	30.0000i		−	0.988534i
$922$		0			0
$923$	−24.0000	+	36.0000i	−0.789970	+	1.18495i
$924$		0			0
$925$			8.00000i			0.263038i
$926$		0			0
$927$	−8.00000			−0.262754
$928$		0			0
$929$		−	6.00000i		−	0.196854i	−0.995144	−	0.0984268i	$-0.968619\pi$
$929$		−	6.00000i		−	0.196854i	0.995144	−	0.0984268i	$-0.0313810\pi$
$930$		0			0
$931$		−	6.00000i		−	0.196642i
$932$		0			0
$933$	−12.0000			−0.392862
$934$		0			0
$935$	48.0000			1.56977
$936$		0			0
$937$	−2.00000			−0.0653372			−0.0326686	−	0.999466i	$-0.510401\pi$
$937$	−2.00000			−0.0653372			−0.0326686	+	0.999466i	$0.510401\pi$
$938$		0			0
$939$	22.0000			0.717943
$940$		0			0
$941$			10.0000i			0.325991i	0.986627	+	0.162995i	$0.0521156\pi$
$941$			10.0000i			0.325991i	−0.986627	+	0.162995i	$0.947884\pi$
$942$		0			0
$943$		−	24.0000i		−	0.781548i
$944$		0			0
$945$	−4.00000			−0.130120
$946$		0			0
$947$			48.0000i			1.55979i	0.625910	+	0.779895i	$0.284728\pi$
$947$			48.0000i			1.55979i	−0.625910	+	0.779895i	$0.715272\pi$
$948$		0			0
$949$	8.00000	−	12.0000i	0.259691	−	0.389536i
$950$		0			0
$951$		−	18.0000i		−	0.583690i
$952$		0			0
$953$	−6.00000			−0.194359			−0.0971795	−	0.995267i	$-0.530982\pi$
$953$	−6.00000			−0.194359			−0.0971795	+	0.995267i	$0.530982\pi$
$954$		0			0
$955$			40.0000i			1.29437i
$956$		0			0
$957$		−	24.0000i		−	0.775810i
$958$		0			0
$959$	28.0000			0.904167
$960$		0			0
$961$	27.0000			0.870968
$962$		0			0
$963$	−16.0000			−0.515593
$964$		0			0
$965$	−32.0000			−1.03012
$966$		0			0
$967$			2.00000i			0.0643157i	0.999483	+	0.0321578i	$0.0102379\pi$
$967$			2.00000i			0.0643157i	−0.999483	+	0.0321578i	$0.989762\pi$
$968$		0			0
$969$		−	12.0000i		−	0.385496i
$970$		0			0
$971$	−4.00000			−0.128366			−0.0641831	−	0.997938i	$-0.520444\pi$
$971$	−4.00000			−0.128366			−0.0641831	+	0.997938i	$0.520444\pi$
$972$		0			0
$973$		−	40.0000i		−	1.28234i
$974$		0			0
$975$	3.00000	+	2.00000i	0.0960769	+	0.0640513i
$976$		0			0
$977$		−	42.0000i		−	1.34370i	−0.740688	−	0.671850i	$-0.765500\pi$
$977$		−	42.0000i		−	1.34370i	0.740688	−	0.671850i	$-0.234500\pi$
$978$		0			0
$979$	24.0000			0.767043
$980$		0			0
$981$		−	20.0000i		−	0.638551i
$982$		0			0
$983$			4.00000i			0.127580i	0.997963	+	0.0637901i	$0.0203188\pi$
$983$			4.00000i			0.127580i	−0.997963	+	0.0637901i	$0.979681\pi$
$984$		0			0
$985$	12.0000			0.382352
$986$		0			0
$987$	−16.0000			−0.509286
$988$		0			0
$989$	−16.0000			−0.508770
$990$		0			0
$991$	−16.0000			−0.508257			−0.254128	−	0.967170i	$-0.581789\pi$
$991$	−16.0000			−0.508257			−0.254128	+	0.967170i	$0.581789\pi$
$992$		0			0
$993$		−	6.00000i		−	0.190404i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	42.0000			1.33015			0.665077	−	0.746775i	$-0.268399\pi$
$997$	42.0000			1.33015			0.665077	+	0.746775i	$0.268399\pi$
$998$		0			0
$999$			8.00000i			0.253109i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2496.2.c.n.961.1		2
4.3	odd	2		2496.2.c.g.961.1		2
8.3	odd	2		312.2.c.b.25.2	yes	2
8.5	even	2		624.2.c.b.337.2		2
13.12	even	2	inner	2496.2.c.n.961.2		2
24.5	odd	2		1872.2.c.a.1585.1		2
24.11	even	2		936.2.c.a.649.1		2
52.51	odd	2		2496.2.c.g.961.2		2
104.5	odd	4		8112.2.a.k.1.1		1
104.21	odd	4		8112.2.a.e.1.1		1
104.51	odd	2		312.2.c.b.25.1	✓	2
104.77	even	2		624.2.c.b.337.1		2
104.83	even	4		4056.2.a.p.1.1		1
104.99	even	4		4056.2.a.n.1.1		1
312.77	odd	2		1872.2.c.a.1585.2		2
312.155	even	2		936.2.c.a.649.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
312.2.c.b.25.1	✓	2	104.51	odd	2
312.2.c.b.25.2	yes	2	8.3	odd	2
624.2.c.b.337.1		2	104.77	even	2
624.2.c.b.337.2		2	8.5	even	2
936.2.c.a.649.1		2	24.11	even	2
936.2.c.a.649.2		2	312.155	even	2
1872.2.c.a.1585.1		2	24.5	odd	2
1872.2.c.a.1585.2		2	312.77	odd	2
2496.2.c.g.961.1		2	4.3	odd	2
2496.2.c.g.961.2		2	52.51	odd	2
2496.2.c.n.961.1		2	1.1	even	1	trivial
2496.2.c.n.961.2		2	13.12	even	2	inner
4056.2.a.n.1.1		1	104.99	even	4
4056.2.a.p.1.1		1	104.83	even	4
8112.2.a.e.1.1		1	104.21	odd	4
8112.2.a.k.1.1		1	104.5	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2496 = 2^{6} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2496.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -2.00000i q^{5} -2.00000i q^{7} +1.00000 q^{9} +4.00000i q^{11} +(3.00000 + 2.00000i) q^{13} -2.00000i q^{15} +6.00000 q^{17} -2.00000i q^{19} -2.00000i q^{21} +4.00000 q^{23} +1.00000 q^{25} +1.00000 q^{27} -6.00000 q^{29} +2.00000i q^{31} +4.00000i q^{33} -4.00000 q^{35} +8.00000i q^{37} +(3.00000 + 2.00000i) q^{39} -6.00000i q^{41} -4.00000 q^{43} -2.00000i q^{45} -8.00000i q^{47} +3.00000 q^{49} +6.00000 q^{51} -2.00000 q^{53} +8.00000 q^{55} -2.00000i q^{57} +8.00000i q^{59} +14.0000 q^{61} -2.00000i q^{63} +(4.00000 - 6.00000i) q^{65} -14.0000i q^{67} +4.00000 q^{69} +12.0000i q^{71} -4.00000i q^{73} +1.00000 q^{75} +8.00000 q^{77} +1.00000 q^{81} -12.0000i q^{85} -6.00000 q^{87} -6.00000i q^{89} +(4.00000 - 6.00000i) q^{91} +2.00000i q^{93} -4.00000 q^{95} -12.0000i q^{97} +4.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + 2 q^{9} + 6 q^{13} + 12 q^{17} + 8 q^{23} + 2 q^{25} + 2 q^{27} - 12 q^{29} - 8 q^{35} + 6 q^{39} - 8 q^{43} + 6 q^{49} + 12 q^{51} - 4 q^{53} + 16 q^{55} + 28 q^{61} + 8 q^{65} + 8 q^{69}+ \cdots - 8 q^{95}+O(q^{100}) \) 2 * q + 2 * q^3 + 2 * q^9 + 6 * q^13 + 12 * q^17 + 8 * q^23 + 2 * q^25 + 2 * q^27 - 12 * q^29 - 8 * q^35 + 6 * q^39 - 8 * q^43 + 6 * q^49 + 12 * q^51 - 4 * q^53 + 16 * q^55 + 28 * q^61 + 8 * q^65 + 8 * q^69 + 2 * q^75 + 16 * q^77 + 2 * q^81 - 12 * q^87 + 8 * q^91 - 8 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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